The ultimate goal of the proposed Phase I and Phase II is to develop advanced computational tools, based on finite element and fast hierarchical multipole methods, for the realistic, large-scale numerical simulation of polyelectrolyte DNA supercoils using high-performance parallel-processing supercomputers. DNA supercoiling is an important facet of biological processes that entail local helical unwinding or denaturation. Recent computer simulations have demonstrated the potential for clarifying the biological role of local structures and ligand binding on DNA super- coiling. The polyelectrolyte character of the DNA has not been included because of computational constraint. The present effort aims to develop a fast O(NlogN) method for the robust treatment of the polyelectrolyte effect in prototypical Monte Carlo and finite element computations. The method will be implemented on advanced parallel- processing platforms using a novel low-overhead communication strategy. The new strategy exploits the multi-level near-field/far-field structure of the fast hierarchical method to eliminate the globality of the electrostatic problem. The resulting computational tools will be designed to take full advantage of the computing power put forth by high- performance parallel-processing supercomputers and will serve to showcase a Grand Challenge application of High Performance Computing in largescale biological systems.